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A recent protocol innovation with magnetic resonance imaging (MRI) has resulted in diffusion tensor imaging (DTI). The approach holds tremendous promise for improving our understanding of neural pathways, especially in the brain. MRIs work by recording displacements at a molecular level. The DTI protocol highlights the distribution of water molecules (in three dimensions). In a medium with free water motion, the diffusion of water molecules is expected to be isotropic, the same in all directions. With water embedded innonhomogeneous tissue, motion is expected to be anisotropic, not the same in all directions, and might show preferred directions of mobility. DTI fully characterizes diffusion anisotropy locally in space, thus providing rich detail about tissue microstructure. However, little has been done to define metrics or describe credible statistical methods for analyzing DTI data. This dissertation will show that the Geisser-Greenhouse sphericity estimator can be approximated by a squared beta distribution. Noise will also be added to show these fits also work for simulated diffusion tensors. Diagnostics are extremely important prior to analyzing these data. There are various regions, especially in the brain, where the distribution of the fractional anisotropy values could be bimodal. This is most likely due to partial voluming affects in imaging, where a voxel (volume of space) may incorporate more than the region of interest. However, the bimodal distribution can also be the result of picking up both white and grey matter in the region. If checks are not done prior to the analysis, all the results may be incorrect, since the main assumption (approximate F) would not be valid. By using diagnostic approaches like QQ-envelop and SiZer, one can examine whether the approximations are reasonable. If appropriate, the methodology previously discussed can be used. However, if these approximations do not apply, new methods will be necessary to analyze the data. Different methods for analyzing the data will be considered, these methods will include: finding an approximate bimodal distribution and the DiProPerm (Direction PROjection PERMutation) test.